Approximating definite integrals Use a Taylor series to approximate the following definite integrals. Retain as many terms as needed to ensure the error is less than 10⁻⁴.∫₀⁰ᐧ²⁵ e⁻ˣ² dx

Verified step by step guidance

1. Recognize that the integral involves the function \(f(x) = e^{-x^2}\), which does not have an elementary antiderivative, making it a good candidate for approximation using a Taylor series expansion.
2. Write the Taylor series expansion of \(e^{-x^2}\) centered at \(x=0\). Recall that the exponential function \(e^u\) can be expanded as \(\sum_{n=0}^{\infty} \frac{u^n}{n!}\). Here, \(u = -x^2\), so the series becomes \(e^{-x^2} = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n}}{n!}\).
3. Substitute the Taylor series into the integral to approximate it: \(\int_0^{0.25} e^{-x^2} \, dx = \int_0^{0.25} \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n}}{n!} \, dx\). Interchange the integral and the summation (justified by uniform convergence on the interval) to get \(\sum_{n=0}^{\infty} \frac{(-1)^n}{n!} \int_0^{0.25} x^{2n} \, dx\).
4. Evaluate the integral inside the summation: \(\int_0^{0.25} x^{2n} \, dx = \frac{(0.25)^{2n+1}}{2n+1}\). So the approximation becomes \(\sum_{n=0}^{\infty} \frac{(-1)^n}{n!} \cdot \frac{(0.25)^{2n+1}}{2n+1}\).
5. Determine how many terms to retain by estimating the remainder term to ensure the error is less than \$10^{-4}\(. Use the alternating series remainder estimation or compare the magnitude of the next term to \)10^{-4}$. Stop adding terms once the next term's absolute value is smaller than this threshold.

Key Concepts

Taylor Series Expansion

A Taylor series represents a function as an infinite sum of terms calculated from the function's derivatives at a single point. For e^(-x²), expanding around x=0 allows approximation by polynomials, simplifying integration. Retaining enough terms ensures the approximation meets the desired accuracy.

Definite Integral Approximation Using Series

Integrating a function approximated by a Taylor series term-by-term converts the integral into a sum of integrals of polynomials. This method simplifies evaluating definite integrals that lack elementary antiderivatives, like ∫₀^{0.25} e^{-x²} dx.

Error Estimation and Convergence Criteria

To ensure the approximation error is below 10⁻⁴, one must estimate the remainder term of the Taylor series. This involves bounding the next term's magnitude or using known error formulas, guiding how many terms to retain for the integral approximation.